# Theory

## Surface Crystallography

The surface of a solid is a defect - it breaks translational symmetry of the lattice. It is a region in which bulk solid physics is no longer adequate, resulting in the development of a branch of physics solely to understand the surface. Although the surface destroys the 3D periodicity of the solid, 2D periodicity remains in the direction parallel to the surface. As in the case of 3D solids, the requirement of 2D surface periodicity results in a finite number of possible unit lattices, described by the 5 2D Bravais nets (Hook and Hall 1991, 7).

In the case of an ideal surface, the solid is cut along a certain direction to reveal the 2D projection of the 3D structure (see Figure 1). Figure 1 – The BCC unit cell (left) showing the (011) plane. The image on the right is a bird’s eye view of the crystal cut along the (011) plane. The darker atoms make up the 2nd layer of atoms.

In the real case, however, the surface undergoes varying degrees of relaxation or reconstruction in order to minimise the free energy available. This breaks the periodicity perpendicular to the surface, in a few atomic layers. This region is known as the selvedge. The substrate is found beneath the selvedge, in which the perpendicular periodicity returns. Reconstruction can occur due to reactions between atoms already present on the surface, or between surface atoms and atoms from the surrounding environment (adatoms). There are numerous ways to describe the surface reconstruction however in this report, I shall use the Wood notation, which is the focus of the next sub-section.

## Wood Notation

In the Wood notation, the reconstructed surface unit mesh (b1, b2) is described in terms of the substrate surface unit mesh (a1, a2). Let,

then the surface lattice is defined as an (m x n) reconstruction with relation to the substrate net. If the surface lattice parameters are rotated by an angle θ relative to the substrate an Rθ is added after the brackets. Figure 2 gives some examples of this notation. Figure 2 – Different unit cell reconstructions 1) (2x1), 2) (1x2) 3) (2x2) and 4) (√2x√2)R45 for the square substrate unit cell described by a1 and a2.

## Electron Counting

It is possible to guess the reconstruction a surface will undergo using a theoretical model, the electron counting model. When a surface is created, bonds are broken, leaving behind unpaired bonding electrons known as dangling bonds. In electronegative atoms the dangling bond electrons will be in the valence energy band where as for electropositive atoms the dangling bond electrons will be in the conduction energy band. The electron counting model requires that the number of available electrons in the surface will exactly fill all the valence band dangling bonds, leaving conduction band dangling bonds empty. A surface that satisfies this condition is semiconducting, whereas one that does not, and has partially filled conduction band dangling bonds, is said to be metallic (M. D. Pashley 1989).